Parallel transport and layer-resolved thermodynamic measurements in twisted bilayer graphene

We employ dual-gated 30{\deg}-twisted bilayer graphene to demonstrate simultaneous ultra-high mobility and conductivity (up to 40 mS at room temperature), unattainable in a single-layer of graphene. We find quantitative agreement with a simple phenomenology of parallel conduction between two pristine graphene sheets, with a gate-controlled carrier distribution. Based on the parallel transport mechanism, we then introduce a method for in situ measurements of the chemical potential of the two layers. This twist-enabled approach, neither requiring a dielectric spacer, nor separate contacting, has the potential to greatly simplify the measurement of thermodynamic quantities in graphene-based systems of high current interest.

2 present transport measurements on a dual-gated 30°-twisted bilayer graphene (30TBG) device encapsulated in hexagonal Boron Nitride (hBN). Thanks to low-energy interlayer decoupling and high device quality, we show that 30TBG replicates the transport properties of two pristine SLG sheets conducting in parallel, including ultra-high mobility at large carrier concentration.
Furthermore, we show that dual-gated 30TBG enables in situ -or, better said, in devicethermodynamic measurements on the individual graphene layers. Measuring the chemical potential (μ) as a function of experimental knobs such as n or an external magnetic field (B) can greatly contribute to the understanding of novel two-dimensional systems. To this end, probes such as scanning single-electron transistors have been widely employed on graphene [13,14]. Experiments based on capacitance spectroscopy can be performed using either metallic [15,16] or graphitic gates [17]. Kim et al. introduced a double-SLG configuration, allowing reciprocal measurements of μ(n,B) via dc electrical transport [18]. This technique, however, requires a dielectric spacer (typically few-nanometer thick hBN) to ensure capacitive coupling between separately-contacted graphene sheets [19,20,21]. Here, we exploit the effective electronic decoupling of large-angle TBG to obtain equivalent information via dc electrical transport in a standard dual-gated device. In particular, by keeping one of the layers charge-neutral, it is possible to probe μ in the other one with a resolution in the meV range (comparable to hBN-spaced structures [20]), as demonstrated by measurements of the SLG Landau level (LL) energies at B = 250 mT.
30TBG is synthesized via chemical vapor deposition (CVD) on Cu, encapsulated in hBN and processed into a dual-gated device (see sketch in Fig. 1(a) and Supplemental Material [22] for details), using the methods developed in Ref. [23]. The selective growth of 30TBG on Cu is favored by the interaction between step edges on the Cu surface and the graphene layers, which triggers the orientation of straight (either zigzag or armchair) graphene edges along the step direction [24]. Transfer of 30TBG to SiO2/Si and hBN-mediated pickup are performed while always keeping T < 170 °C to avoid possible relaxation of the twist angle. The stability of the 30°-rotated configuration upon hBN encapsulation was confirmed by transmission electron microscope selected area electron diffraction, as reported in Ref. [23]. The assembly technique by Purdie et al. [25] is used to promote interface cleaning and obtain high-quality electronic transport.
The top-and back-gate voltages (Vtg and Vbg) couple to TBG via hBN (32 nm thickness, determined by atomic force microscopy) and in-series SiO2-hBN (285-40 nm thickness), respectively, resulting in the capacitance-per-unit-area Ctg = 8.3×10 -8 F/cm 2 and Cbg = 9.8×10 -9 F/cm 2 . In addition, it is well established that a considerable interlayer capacitance Cgg = 7.5×10 -6 F/cm 2 must be considered for large-angle TBG [26,27]. Data in Figure 1(b) show the longitudinal resistivity of 30TBG measured at room T, as a function of Vtg and Vbg. The high-resistivity diagonal (ρ up to ~400 Ω) corresponds to the global charge neutrality point (CNP) of the sample, while ρ as low as 25 Ω is measured at large carrier concentration. The fact that the global CNP crosses exactly Vtg = Vbg = 0 V indicates negligible residual doping in the device. As shown by the curves in Fig. 1(c) inset, the resistivity peak becomes wider and shallower upon unbalancing Vtg and Vbg, i.e., by increasing the so-called displacement electric field (D). This is caused by a splitting of the CNPs for the two layers [27], although not resolvable at room T due to thermal broadening. To extract quantitative information from the transport data, it is necessary to have knowledge of the individual carrier density in the layers (nupper and nlower), which can be obtained by electrostatic modelling of the gated TBG system [26,27]; complete details on our procedure are provided in Supplemental Material [22].  Figure 1(b), which correspond to fixed differences in the gate potentials (weighted with their respective gate capacitance). We note that such trajectories do not coincide with constant values of displacement field (as defined in the electrostatic model in Supplemental Material [22]), unless at D = 0 (black line in Figure 1b). While at D = 0 the layers are charge-balanced (black line, nupper = nlower), increasingly separated and nonlinear Vtg-dependences are observed otherwise. Complete color plots of the Vtg-Vbg dependence of nupper and nlower are shown in Fig. S1(a) and S1(b) in Supplemental Material [22].
Based on such relations, we can calculate the total carrier density in the system (ntot = nupper + nlower) and use the standard Drude model to convert the experimental ρ data into the mobility | | shown in Fig.   1(d). At ntot < 10 12 cm -2 , we observe a carrier mobility as high as 1.9×10 5 cm 2 V -1 s -1 , and a weak dependence on the interlayer charge configuration, ascribable to the D-induced broadening of the resistivity peak. However, such difference becomes irrelevant at ntot > 10 12 cm -2 , where the four experimental curves collapse on each other. Here, the room T mobility of 30TBG strikingly surpasses the intrinsic phononlimited behavior of SLG (blue dotted line) [28,29], although falling below the theoretical limit for two parallel-conducting SLG for ntot < 4×10 12 cm -2 (blue solid line, calculated assuming nupper = nlower and a density-independent resistivity of 52 Ω in each layer [28]). This fact can be reasonably expected based on experiments on hBN-encapsulated SLG, where a room T mobility below the phonon limit is observed at relatively low carrier density (n < 2×10 12 cm -2 ) [30]. For ntot > 4×10 12 cm -2 , 30TBG perfectly mimics two phonon-limited SLG that simply conduct in parallel, leading to an unprecedented room T mobility at large carrier density (~5×10 4 cm 2 V -1 s -1 at ntot ~ 5×10 12 cm -2 , corresponding to σ ~ 40 mS). Such combination might have important applicative implications for high-speed electronics [7,[31][32][33], integrated optoelectronics [34,35], power conversion efficiency in solar cells [36,37], and sensing [38,39]. 4 In Fig. 2, we further investigate the parallel transport mechanism in TBG at low temperature (T = 4.2 K).
We again consider several trajectories at fixed gate difference (lines in Fig. 2(a) inset). However, to avoid a small resistive feature attributable to a contact malfunctioning at Vbg ~ 0 V (Vtg-independent feature in Fig. 2(a) inset, see also Fig. S2 in Supplemental Material [22]), we consider half of the ρ curves from the lower left quadrant (for ntot < 0), half from the upper right one (for ntot > 0). At low displacement field, the resistivity shows a sharp peak centered at ntot = 0, confirming the high quality of the CVD-grown crystals and the low disorder in the hBN-encapsulated device ( Fig. 2(a), see also discussion in Supplemental Material [22]). With increasing D, strong broadening and attenuation (see horizontal markers in Fig. 2(a)) of the peak are observed. Fig. 2(b) shows a zoom-in for ρ ≤ 60 Ω, which allows evaluating the behavior at large ntot. We find that the 30TBG resistivity perfectly reproduces that of two parallel-conducting graphene layers, in which the carrier mean free path (lmfp) is set by the width of the device channel 2.2 μm, and the conductivity in each SLG is given by = 2 . This indicates predominant scattering at the device boundaries, consistent with results on hBN-encapsulated SLG [30]. The large-ntot limit is independent of the interlayer carrier distribution, consistent with the observations at room T. However, we find considerable effects on the transport properties at low carrier density. In SLG, the n * parameterthe characteristic density at which log(σ) switches from saturated (electron-hole puddles dominated) to linear-in-log(n) (single-carrier type) -is determined by long-range disorder and correlates with the inverse of the carrier mobility [40]. In TBG, we find that large values of ntot * can instead be determined solely by the screening of the applied electric field, and do not necessarily correspond to an increased disorder level. The vertical lines in Fig 2(c)), two parallel-conducting SLG with a disorder-determined ntot * = 10 11 cm -2 (i.e., 5×10 10 cm -2 per layer) are expected to show a mobility ~8×10 4 cm 2 V -1 s -1 according to the 1/n * dependence for SLG reported in Ref. [40]. Such discrepancy should be taken into careful account, especially in experiments on TBG employing a single gate electrode. The quantitative agreement, both for ρ and ntot * , with simple parallel transport indicates that the interlayer conductivity is negligible with respect to that along the constituent layers [41]. At 30° twisting, the suppression of interlayer transport might be further favored by the incommensurate stacking configuration [42].

5
In the following we show the possibility of employing one of the two graphene sheets (the probe) to sense the chemical potential and the carrier concentration in the other one (the target). The key strategy consists in the individuation and tracking of the probe CNP in the parallel transport measurements as a function of the gate potentials. In Fig. 3(a) we plot the low T conductivity as a function of Vtg relative to the global CNP, here defined as ΔVtg = Vtg + Vbg ×Cbg/Ctg , and Vbg. In this zoomed plot, we clearly observe a nonlinear separation between the upper and lower CNPs, resulting from the interlayer capacitive coupling [27]. The CNPs form an hourglass-like shape, separating regions with equal (e-e and h-h, high σ) and opposite carrier sign (e-h and h-e, low σ) on the two layers. The e-h coexistence, expected from calculations of nupper and nlower (see Fig. S1(c) and S1(d) in Supplemental Material [22]), is demonstrated by magnetotransport measurements at Vbg = -40 V shown in Fig. S3 in Supplemental Material [22]. We note that the lower conductivity observed in the e-h configuration is due to the smaller total carrier density and its reduced gate-dependence (see Fig. S4 in Supplemental Material [22]). The position of the probe CNP can be accurately determined by considering log(σ) as a function of log(ΔVtg) along horizontal cuts from Fig. 3(a), and applying a procedure similar to that commonly used to extract n * in SLG. As shown in In principle, our approach can be used to obtain high-resolution thermodynamic information on an arbitrary graphene-based system (for instance, magic-angle TBG as in Ref. [20]), simply by stacking it on top of SLG and imposing a large relative twisting. We stress that the 30° rotation is not a strict requirement: smaller twist angles could be equivalently employed, as long as the interlayer decoupling is preserved [26,27]. While in the case presented here of SLG probing SLG, the value of the interlayer capacitance Cgg is well established, this might vary significantly when considering a different target system.
However, given the reciprocity of the technique, Cgg can be obtained by measuring μ(n) of the probe SLG via tracking the target CNP, initializing a reasonable starting value of interlayer capacitance (e.g that of large-angle TBG) and iterating the procedure until convergence to the SLG Dirac dependence. We note that both the parallel transport mechanism and the thermodynamic measurement scheme could be extended to other van der Waals systems where interlayer decoupling is obtained from large-angle twisting, such as twisted transition metal dichalcogenides.
In summary, we have shown that it is possible to achieve simultaneous ultra-high mobility and conductivity in 30TBG, based on parallel conduction and gate-controlled carrier distribution. In addition, we have exploited the low-energy electronic decoupling of 30TBG to introduce a technique for layersensitive thermodynamic measurements using a standard device structure and measurement configuration.     fabricated, avoiding short circuit with the electrical contacts and with exposed TBG edges. Finally, the heterostructure is etched (same recipe as above) in a 6-terminal Hall bar shape, using the physical mask provided by the metallic top gate, in combination with additional e-beam-patterned PMMA arms. The Hall bar is 2.2 μm wide, the distance between the voltage probes is 3 μm.
The sample is mounted in a variable-temperature insert, loaded into a liquid-helium cryostat with superconducting coil. All the measurements are performed in four-probe configuration, with constant current excitation (10-100 nA) and standard lock-in acquisition (13 Hz).

Electrostatic model
Modelling the electrostatics of dual-gated TBG allows obtaining the individual carrier density in the two layers (nupper and nlower) at arbitrary values of the applied gate voltages (Vtg and Vbg). Sanchez-Yamagishi et al. [1] firstly reported on the screening properties of large-angle TBG, and pointed out the importance of including the chemical potential of the individual layers in the electrostatic description (see SM file therein [1]). Equivalent approaches, although with different naming of the relevant quantities, were used more recently by Park et al. [2] (hBN-spaced double-layer), and Rickhaus et al. [3] (large-angle TBG, with and without hBN spacer). Following the definitions of Ref. [1], the total carrier density and electric displacement field are given by where Vtg and Vbg are the applied gate voltages, and the chemical potential Ref. [3]), which determines a chemical potential difference Δ = − [1]. Therefore, the displacement field in Equation S2 can be split into two components, according to To determine nupper as a function of Vtg and Vbg, we use the following procedure (equivalently for nlower): 1. Take a dense sequence of equally-spaced values of ntot over an experimentally relevant range, e.g.
[-4.8×10 12  7. Go back to step 2 and iterate the procedure for arbitrary values of nupper.
Our complete results are shown in Figure S1(a) (nupper) and Figure S1(b) (nlower). The same procedure can be repeated to obtain the layers' carrier density in the zoomed-in ΔVtg-Vbg space ( Figure S1(c) and S1(d)).
Based on these plots, we can express the experimental data (obtained as a function of the gate potentials along a given Vtg-Vbg trajectory) as a function of the total carrier density (see main text Figure 1 and Figure   2). We use an algorithm that calculates the voltage distances between each point of the chosen experimental trajectory and all the points in both Figures S1(a) and (b), and then finds the minimum distances. Next, the algorithm associates to each point of the experimental trajectory a total charge density given by the sum of nupper and nlower corresponding to the minimum distance point.

Low temperature resistivity curves
The low T resistivity curves in main text Figure 2(a) are plotted considering half of the ρ data from the lower left quadrant (ntot < 0) of the Vtg-Vbg space, half from the upper right one (ntot > 0). This is done to avoid a small resistive peak independent from Vtg, and located at Vbg ~ 0 V, shown in Figure S2. The insensitivity to Vtg indicates that this feature originates from the contact areas, which are not covered by the top-gate. At Vbg ~ 0 V, the TBG close to the contacts is undoped and minor malfunctioning is likely.

Extrinsic disorder in CVD-grown 30TBG
The use CVD-grown 30TBG crystals is mainly motivated by the capability of avoiding the van der Waals assembly step and obtaining TBG with high yield (tens of samples from a single growth cycle) [4]. However, in analogy to recent reports on CVD-grown SLG [5,6], such choice does not harm the quality of the electrical transport when combined with dry hBN encapsulation. The following observations from low T experiments support this claim: (i) from the width of the CNP region (n*tot) at low displacement field, the charge fluctuations in our device are ~10 10 cm -2 , which corresponds to ~5×10 9 cm -2 in each graphene layer. Such magnitude is fully comparable with ultra-clean devices based on exfoliated flakes [7].
(ii) the carrier mean free path in each layer is ~2.2 μm (see Figure 2b), which equals the physical width of the device channel. Hence, the main scattering contribution can be identified with the sample edges, again matching the results of the community on exfoliation-based devices [8].
(iii) a complete set of interlayer quantum Hall states is resolved at B = 250 mT ( Figure 3d) and quantum oscillations start to be visible at ~50 mT ( Figure S3 in SM), indicating low disorder broadening of the LLs [9].
e-h coexistence at high displacement field In order to prove the presence of charge carriers of opposite sign, we perform magnetotransport at high displacement field. In Figure S3 we Gate-dependent total carrier density in the e-h configuration In main text Figure 3(a), the low T conductivity shows regions of different magnitude corresponding to different charge configurations, separated by the CNPs of the two layers. The conductivity in the e-h (and h-e) region is found to be lower with respect to the case of equal carrier sign. Despite the fact that both layers have a finite carrier density, unless at ΔVtg = Vbg = 0, the absolute value of the total carrier density |ntot|=|nupper|+ |nlower| keeps relatively low in the e-h region, therefore limiting the measured TBG conductivity. From the maps in Figure S1(c) and S1(d), we can obtain |ntot| along different trajectories in the ΔVtg-Vbg space. In Figure S4 we show that an "horizontal" movement along Vbg = 0 (red points, e-e ) yields a strong linear increase in |ntot|, leading to large values of σ. Instead, a "vertical" movement along ΔVtg= 0 (black points, h-e) induces less |ntot|, with a pronounced sublinear dependence, responsible for the lower σ measured experimentally.